Final answer:
The partial fraction decomposition of the given rational expression is obtained by setting up an equation with unknown coefficients, multiplying through by the common denominator, and solving for the coefficients to express the original expression as a sum of simpler fractions.
Step-by-step explanation:
To find the partial fraction decomposition of the given rational expression (6x²+11x+15)/(x+1)(x²+x+5), we first assume that it can be represented as the sum of fractions where:
A/(x+1) + (Bx+C)/(x²+x+5) = (6x²+11x+15)/(x+1)(x²+x+5)
Now, we multiply both sides of the equation by the common denominator (x+1)(x²+x+5) to get rid of the denominators:
A(x²+x+5) + (Bx+C)(x+1) = 6x²+11x+15
Then we expand and combine like terms to equate the coefficients of corresponding powers of x on both sides of the equation. This gives us a system of equations which we solve for A, B, and C.
Once we have the values of A, B, and C, we substitute them back into the partial fraction form:
A/(x+1) + (Bx+C)/(x²+x+5)
This gives us the complete partial fraction decomposition of the original rational expression.